A056557 -id:A056557 - cp's OEIS frontend

A127321 First 4-dimensional hyper-tetrahedral coordinate; repeat m C(m+3,3) times; 4-D analog of A056556.

0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 0

Graeme McRae, Jan 10 2007

nonn

If {(W,X,Y,Z)} are 4-tuples of nonnegative integers with W>=X>=Y>=Z ordered by W, X, Y and Z, then W=A127321(n), X=A127322(n), Y=A127323(n) and Z=A127324(n). These sequences are the four-dimensional analogs of the three-dimensional A056556, A056557 and A056558.

			a(23)=3 because a(A000332(3+3)) = a(A000332(3+4)-1) = 3, so a(15) = a(34) = 3.
Table of A127321, A127322, A127323, A127324:
  n W,X,Y,Z
  0 0,0,0,0
  1 1,0,0,0
  2 1,1,0,0
  3 1,1,1,0
  4 1,1,1,1
  5 2,0,0,0
  6 2,1,0,0
  7 2,1,1,0
  8 2,1,1,1
  9 2,2,0,0
 10 2,2,1,0
 11 2,2,1,1
 12 2,2,2,0
 13 2,2,2,1
 14 2,2,2,2
 15 3,0,0,0
 16 3,1,0,0
 17 3,1,1,0
 18 3,1,1,1
 19 3,2,0,0
 20 3,2,1,0
 21 3,2,1,1
 22 3,2,2,0
 23 3,2,2,1

Michael De Vlieger, Table of n, a(n) for n = 0..10625, showing all instances of m=0..21.

Cf. A127322, A127323, A127324, A056556, A056557, A056558, A000332, A000292, A000217.

Array[Floor[Sqrt[5/4 + Sqrt[24*# + 1]] - 3/2] &, 105, 0] (* or *)
Flatten@ Array[ConstantArray[#, Binomial[# + 3, 3]] &, 6, 0] (* Michael De Vlieger, Oct 21 2021 *)

from math import comb
from sympy import integer_nthroot
def A127321(n): return (m:=integer_nthroot(24*(n+2),4)[0]-2)+(n>=comb(m+4,4)) # Chai Wah Wu, Nov 04 2024

For W>=0, a(A000332(W+3)) = a(A000332(W+4)-1) = W A127321(n+1) = A127321(n)==A127324(n) ? A127321(n)+1 : A127321(n).
a(n) = floor(sqrt(5/4 + sqrt(24*n+1)) - 3/2). - Ridouane Oudra, Oct 21 2021
a(n) = m-2 if nChai Wah Wu, Nov 04 2024

Name corrected by Ridouane Oudra, Oct 21 2021

A331195 Three-column table read by rows: triples (i,j,k) in order sorted from the left.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 2, 0, 0, 2, 1, 0, 2, 1, 1, 2, 2, 0, 2, 2, 1, 2, 2, 2, 3, 0, 0, 3, 1, 0, 3, 1, 1, 3, 2, 0, 3, 2, 1, 3, 2, 2, 3, 3, 0, 3, 3, 1, 3, 3, 2, 3, 3, 3, 4, 0, 0, 4, 1, 0, 4, 1, 1, 4, 2, 0, 4, 2, 1, 4, 2, 2, 4, 3, 0, 4, 3, 1, 4, 3, 2, 4, 3, 3, 4, 4, 0, 4, 4, 1, 4, 4, 2, 4, 4, 3, 4, 4, 4, 5, 0, 0

Offset: 0

Mehmet A. Ates, Jun 08 2020

			For n=[0,1,2] to n=[12,13,14], a[n,n+1,n+2] counts up as such: [0,0,0], [1,0,0], [1,1,0], [1,1,1], [2,0,0], etc.

Mehmet A. Ates, Table of n, a(n) for n = 0..10000

Cf. A056556, A056557, A056558, A330709.

See A372667 for the norms of these triples.

ThreeDVectors = List[];
SeqSize = 10;
For[i = 0, i <= SeqSize, i++,
  For[j = 0, j <= i, j++,
    For[k = 0, k <= j, k++,
      AppendTo[ThreeDVectors, {i, j, k}]
    ]
  ]
];
Flatten[ThreeDVectors]

from math import comb, isqrt
from sympy import integer_nthroot
def A331195(n): return (m:=integer_nthroot((n<<1)+6,3)[0])-(n<3*comb(m+2,3)) if not (a:=n%3) else (k:=isqrt(r:=(b:=n//3)+1-comb((m:=integer_nthroot((n<<1)-1,3)[0])-(b=comb(m+2,3))+1,3))-comb((k:=isqrt(m:=r+1<<1))+(m>k*(k+1)),2) # Chai Wah Wu, Nov 23 2024

a(3*n) = A056556(n).
a(3*n+1) = A056557(n).
a(3*n+2) = A056558(n).

A360010 First part of the n-th weakly decreasing triple of positive integers sorted lexicographically. Each n > 0 is repeated A000217(n) times.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8

Offset: 1

Gus Wiseman, Feb 11 2023

nonn

			Triples begin: (1,1,1), (2,1,1), (2,2,1), (2,2,2), (3,1,1), (3,2,1), (3,2,2), (3,3,1), (3,3,2), (3,3,3), ...

For pairs instead of triples we have A002024.
Positions of first appearances are A050407(n+2) = A000292(n)+1.
The zero-based version is A056556.
The triples have sums A070770.
The second instead of first part is A194848.
The third instead of first part is A333516.
Concatenating all the triples gives A360240.
Cf. A000217, A003056, A056557, A056558, A069905, A158842, A331195.

nn=9;First/@Select[Tuples[Range[nn],3],GreaterEqual@@#&]

from math import comb
from sympy import integer_nthroot
def A360010(n): return (m:=integer_nthroot(6*n,3)[0])+(n>comb(m+2,3)) # Chai Wah Wu, Nov 04 2024

a(n) = A056556(n) + 1 = A331195(3n) + 1.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 + log(2)/4. - Amiram Eldar, Feb 18 2024
a(n) = m+1 if n>binomial(m+2,3) and a(n) = m otherwise where m = floor((6n)^(1/3)). - Chai Wah Wu, Nov 04 2024

A070770 b + c + d where b >= c >= d >= 0 ordered by b then c then d.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 4, 4, 5, 6, 3, 4, 5, 5, 6, 7, 6, 7, 8, 9, 4, 5, 6, 6, 7, 8, 7, 8, 9, 10, 8, 9, 10, 11, 12, 5, 6, 7, 7, 8, 9, 8, 9, 10, 11, 9, 10, 11, 12, 13, 10, 11, 12, 13, 14, 15, 6, 7, 8, 8, 9, 10, 9, 10, 11, 12, 10, 11, 12, 13, 14, 11, 12, 13, 14, 15, 16, 12, 13, 14, 15, 16, 17, 18, 7

Offset: 0

Henry Bottomley, May 06 2002

			Triangle begins:
  0,
  ;
  1;
  2, 3;
  ;
  2;
  3, 4;
  4, 5, 6;
  ;
  3;
  4, 5,
  5, 6, 7;
  6, 7, 8, 9;
  ;
  4;
  5, 6;
  6, 7,  8;
  7, 8,  9, 10;
  8, 9, 10, 11, 12;
  ;
  ...

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A001477, A051162, A070771, A070772 for similar sequences with different numbers of terms summed.

seq(seq(seq(b+c+d,d=0..c),c=0..b),b=0..10); # Robert Israel, Jun 21 2018

for(x=0,5,for(y=0,x,for(z=0,y,print1(x+y+z", ")))) \\ Charles R Greathouse IV, Sep 17 2015

from math import isqrt, comb
from sympy import integer_nthroot
def A070770(n): return (m:=integer_nthroot(6*(n+1),3)[0])+(a:=n>=comb(m+2,3))+(k:=isqrt(b:=(c:=n+1-comb(m+a+1,3))<<1))-((b<<2)<=(k<<2)*(k+1)+1)+c-2-comb(k+(b>k*(k+1)),2) # Chai Wah Wu, Dec 11 2024

a(n) = A056556(n) + A056557(n) + A056558(n).

A194849 Write n = C(i,3)+C(j,2)+C(k,1) with i>j>k>=0; let L[n] = [i,j,k]; sequence gives list of triples L[n], n >= 0.

Original entry on oeis.org

2, 1, 0, 3, 1, 0, 3, 2, 0, 3, 2, 1, 4, 1, 0, 4, 2, 0, 4, 2, 1, 4, 3, 0, 4, 3, 1, 4, 3, 2, 5, 1, 0, 5, 2, 0, 5, 2, 1, 5, 3, 0, 5, 3, 1, 5, 3, 2, 5, 4, 0, 5, 4, 1, 5, 4, 2, 5, 4, 3, 6, 1, 0, 6, 2, 0, 6, 2, 1, 6, 3, 0, 6, 3, 1, 6, 3, 2, 6, 4, 0, 6, 4, 1, 6, 4, 2, 6, 4, 3, 6, 5, 0, 6, 5, 1, 6, 5, 2, 6, 5, 3, 6, 5, 4, 7, 1, 0, 7, 2, 0, 7, 2, 1, 7, 3, 0, 7, 3, 1, 7, 3, 2, 7, 4

Offset: 0

N. J. A. Sloane, Sep 03 2011

			List of triples begins:
[2, 1, 0]
[3, 1, 0]
[3, 2, 0]
[3, 2, 1]
[4, 1, 0]
[4, 2, 0]
[4, 2, 1]
[4, 3, 0]
[4, 3, 1]
[4, 3, 2]
...

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.3, Eq. (20), p. 360.

The three columns are [A194847, A194848, A056558], or equivalently [A056556+2, A056557+1, A056558]. See A194847 for further information.

A360573 Odd numbers with exactly three zeros in their binary expansion.

Original entry on oeis.org

17, 35, 37, 41, 49, 71, 75, 77, 83, 85, 89, 99, 101, 105, 113, 143, 151, 155, 157, 167, 171, 173, 179, 181, 185, 199, 203, 205, 211, 213, 217, 227, 229, 233, 241, 287, 303, 311, 315, 317, 335, 343, 347, 349, 359, 363, 365, 371, 373, 377, 399, 407, 411, 413

Offset: 1

Bernard Schott, Feb 12 2023

If m is a term then 2*m+1 is another term, since if M is the binary expansion of m, then M.1 where . stands for concatenation is the binary expansion of 2*m+1.
A052996 \ {1,3,8} is a subsequence, since for m >= 3, A052996(m) = 9*2^(m-2) - 1 has 100011..11 with m-2 trailing 1 for binary expansion.
A171389 \ {20} is a subsequence, since for m >= 1, A171389(m) = 21*2^m - 1 has 1010011..11 with m trailing 1 for binary expansion.
A198276 \ {18} is a subsequence, since for m >= 1, A198276(m) = 19*2^m - 1 has 1001011..11 with m trailing 1 for binary expansion.
Binary expansion of a(n) is A360574(n).
{8*a(n), n>0} form a subsequence of A353654 (numbers with three trailing 0 bits and three other 0 bits).
Numbers of the form 2^(a+1) - 2^b - 2^c - 2^d - 1 where a > b > c > d > 0. - Robert Israel, Feb 13 2023

			35_10 = 100011_2, so 35 is a term.

Cf. A005408, A023416, A056557, A333516, A353654, A360010, A360574.
Subsequences: A052996 \ {1,3,8}, A171389 \ {20}, A198276 \ {18}.
Odd numbers with k zeros in their binary expansion: A000225 (k=0), A190620 (k=1), A357773 (k=2), this sequence (k=3).

q:= n-> n::odd and add(1-i, i=Bits[Split](n))=3:
select(q, [$1..575])[];  # Alois P. Heinz, Feb 12 2023
# Alternative:
[seq(seq(seq(seq(2^(a+1) - 2^b - 2^c - 2^d - 1, d = c-1..1,-1), c=b-1..2,-1),b=a-1..3,-1),a=4..12)]; # Robert Israel, Feb 13 2023

Select[Range[1, 500, 2], DigitCount[#, 2, 0] == 3 &] (* Amiram Eldar, Feb 12 2023 *)

isok(m) = (m%2) && #select(x->(x==0), binary(m)) == 3; \\ Michel Marcus, Feb 13 2023

def ok(n): return n&1 and bin(n)[2:].count("0") == 3
print([k for k in range(414) if ok(k)]) # Michael S. Branicky, Feb 12 2023

from itertools import count, islice
from sympy.utilities.iterables import multiset_permutations
def A360573_gen(): # generator of terms
    yield from (int('1'+''.join(d)+'1',2) for l in count(0) for d in  multiset_permutations('000'+'1'*l))
A360573_list = list(islice(A360573_gen(),54)) # Chai Wah Wu, Feb 18 2023

from itertools import combinations, count, islice
def agen(): yield from ((1<Michael S. Branicky, Feb 18 2023

from math import comb, isqrt
from sympy import integer_nthroot
def A056557(n): return (k:=isqrt(r:=n+1-comb((m:=integer_nthroot(6*(n+1), 3)[0])-(nA333516(n): return (r:=n-1-comb((m:=integer_nthroot(6*n, 3)[0])+(n>comb(m+2, 3))+1, 3))-comb((k:=isqrt(m:=r+1<<1))+(m>k*(k+1)), 2)+1
def A360010(n): return (m:=integer_nthroot(6*n, 3)[0])+(n>comb(m+2, 3))
def A360573(n):
    a = (a2:=integer_nthroot(24*n, 4)[0])+(n>comb(a2+2, 4))+3
    j = comb(a-1,4)-n
    b, c, d = A360010(j+1)+2, A056557(j)+2, A333516(j+1)
    return (1<Chai Wah Wu, Dec 18 2024

A023416(a(n)) = 3.

Let a = floor((24n)^(1/4))+4 if n>binomial(floor((24n)^(1/4))+2,4) and a = floor((24n)^(1/4))+3 otherwise. Let j = binomial(a-1,4)-n. Then a(n) = 2^a-1-2^(A360010(j+1)+2)-2^(A056557(j)+2)-2^(A333516(j+1)). - Chai Wah Wu, Dec 18 2024

A360240 Weakly decreasing triples of positive integers sorted lexicographically and concatenated.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 3, 1, 1, 3, 2, 1, 3, 2, 2, 3, 3, 1, 3, 3, 2, 3, 3, 3, 4, 1, 1, 4, 2, 1, 4, 2, 2, 4, 3, 1, 4, 3, 2, 4, 3, 3, 4, 4, 1, 4, 4, 2, 4, 4, 3, 4, 4, 4, 5, 1, 1, 5, 2, 1, 5, 2, 2, 5, 3, 1, 5, 3, 2, 5, 3, 3, 5, 4, 1, 5, 4, 2, 5, 4, 3

Offset: 1

Gus Wiseman, Feb 11 2023

nonn

			Triples begin: (1,1,1), (2,1,1), (2,2,1), (2,2,2), (3,1,1), (3,2,1), (3,2,2), (3,3,1), (3,3,2), (3,3,3), ...

The triples have sums A070770.
Positions of first appearances are A158842.
For pairs instead of triples we have A330709 + 1.
The zero-based version is A331195.
- The first part is A360010 = A056556 + 1.
- The second part is A194848 = A056557 + 1.
- The third part is A333516 = A056558 + 1.
Cf. A002024, A003056, A069905.

nn=9;Join@@Select[Tuples[Range[nn],3],GreaterEqual@@#&]

from math import isqrt, comb
from sympy import integer_nthroot
def A360240(n): return (m:=integer_nthroot((n-1<<1)+6,3)[0])+(n>3*comb(m+2,3)) if (a:=n%3)==1 else (k:=isqrt(r:=(b:=(n-1)//3)+1-comb((m:=integer_nthroot((n-1<<1)-1,3)[0])-(b(k<<2)*(k+1)+1) if a==2 else 1+(r:=(b:=(n-1)//3)-comb((m:=integer_nthroot((n-1<<1)-3,3)[0])+(b>=comb(m+2,3))+1,3))-comb((k:=isqrt(m:=r+1<<1))+(m>k*(k+1)),2) # Chai Wah Wu, Jun 07 2025

a(n) = A331195(n-1) + 1.

A379269 Numbers whose binary representation has exactly three zeros.

Original entry on oeis.org

8, 17, 18, 20, 24, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 71, 75, 77, 78, 83, 85, 86, 89, 90, 92, 99, 101, 102, 105, 106, 108, 113, 114, 116, 120, 143, 151, 155, 157, 158, 167, 171, 173, 174, 179, 181, 182, 185, 186, 188, 199, 203, 205, 206, 211, 213, 214, 217

Offset: 1

Chai Wah Wu, Dec 19 2024

Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.

Cf. A023416, A056557, A333516, A360010, A360573.

Select[Range[2^8],Count[IntegerDigits[#,2],0]==3&] (* James C. McMahon, Dec 20 2024 *)

from math import comb, isqrt
from sympy import integer_nthroot
def A056557(n): return (k:=isqrt(r:=n+1-comb((m:=integer_nthroot(6*(n+1), 3)[0])-(nA333516(n): return (r:=n-1-comb((m:=integer_nthroot(6*n, 3)[0])+(n>comb(m+2, 3))+1, 3))-comb((k:=isqrt(m:=r+1<<1))+(m>k*(k+1)), 2)+1
def A360010(n): return (m:=integer_nthroot(6*n, 3)[0])+(n>comb(m+2, 3))
def A379269(n):
    a = (a2:=integer_nthroot(24*n, 4)[0])+(n>comb(a2+2, 4))+2
    j = comb(a,4)-n
    b, c, d = A360010(j+1)+1, A056557(j)+1, A333516(j+1)-1
    return (1<

a(n) = (A360573(n)-1)/2.
A023416(a(n)) = 3.
Let a = floor((24n)^(1/4))+3 if n>binomial(floor((24n)^(1/4))+2,4) and a = floor((24n)^(1/4))+2 otherwise. Let j = binomial(a,4)-n. Then a(n) = 2^a-1-2^(A360010(j+1)+1)-2^(A056557(j)+1)-2^(A333516(j+1)-1).
Sum_{n>=1} 1/a(n) = 1.3949930090659130972172214185888677947877214389482588641632435250211546702139813215203065255971026537... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Dec 21 2024

A332616 a(n) = value of the cubic form A^3 + B^3 + C^3 - 3ABC evaluated at row n of the table in A331195.

Original entry on oeis.org

0, 1, 2, 0, 8, 9, 4, 16, 5, 0, 27, 28, 20, 35, 18, 7, 54, 28, 8, 0, 64, 65, 54, 72, 49, 32, 91, 56, 27, 10, 128, 81, 40, 11, 0, 125, 126, 112, 133, 104, 81, 152, 108, 70, 44, 189, 130, 77, 36, 13, 250, 176, 108, 52, 14, 0, 216, 217, 200, 224, 189, 160, 243

Offset: 0

Mehmet A. Ates, Jun 08 2020

No term in the sequence is congruent to 3 or 6 (mod 9).

			For n=3, a(n) = f[1,1,0] = 1^3 + 1^3 + 0^3 - 3*1*1*0 = 2.

Mehmet A. Ates, Table of n, a(n) for n = 0..19599
Mathematical Association of America, 2019 William Lowell Putnam Mathematical Competition Problems

Cf. A056556, A056557, A056558.
Cf. A330013, A331195.
Cf. A074232 (in ascending order, strictly positive & without duplicates).

SeqSize = 30;
ListSize = 120;
F3List = List[];
f3[a_, b_, c_] := a^3 + b^3 + c^3 - 3*a*b*c
For[i = 0, i <= SeqSize, i++, For[j = 0, j <= i, j++, For[k = 0, k <= j, k++, AppendTo[F3List, f3[i, j, k]]]]]
ListPlot[F3List, PlotLabel -> "a(n)"]
Print["First ", ListSize, " elements of a(n): ", Take[F3List, ListSize]]

a(n) = A056556(n)^3 + A056557(n)^3 + A056558(n)^3 - 3*A056556(n)*A056557(n)*A056558(n).

Edited by N. J. A. Sloane, Aug 06 2020

A379270 Numbers with only digits "1" and three digits "0".

Original entry on oeis.org

1000, 10001, 10010, 10100, 11000, 100011, 100101, 100110, 101001, 101010, 101100, 110001, 110010, 110100, 111000, 1000111, 1001011, 1001101, 1001110, 1010011, 1010101, 1010110, 1011001, 1011010, 1011100, 1100011, 1100101, 1100110, 1101001, 1101010, 1101100

Offset: 1

Chai Wah Wu, Dec 19 2024

Binary representation of A379269.

Numbers in A007088 with three 0 digits.

Cf. A007088, A379269, A023416, A056557, A333516, A360010, A360573.

Select[Range[10^7],Count[IntegerDigits[#],0]==3&&Max[IntegerDigits[#]]==1&] (* James C. McMahon, Dec 20 2024 *)

from math import isqrt, comb
from sympy import integer_nthroot
def A056557(n): return (k:=isqrt(r:=n+1-comb((m:=integer_nthroot(6*(n+1), 3)[0])-(nA333516(n): return (r:=n-1-comb((m:=integer_nthroot(6*n, 3)[0])+(n>comb(m+2, 3))+1, 3))-comb((k:=isqrt(m:=r+1<<1))+(m>k*(k+1)), 2)+1
def A360010(n): return (m:=integer_nthroot(6*n, 3)[0])+(n>comb(m+2, 3))
def A379270(n):
    a = (a2:=integer_nthroot(24*n, 4)[0])+(n>comb(a2+2, 4))+2
    j = comb(a,4)-n
    b, c, d = A360010(j+1)+1, A056557(j)+1, A333516(j+1)-1
    return (10**a-1)//9-10**b-10**c-10**d

a(n) = A007088(A379269(n)).

cp's OEIS Frontend

A127321 First 4-dimensional hyper-tetrahedral coordinate; repeat m C(m+3,3) times; 4-D analog of A056556.

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A331195 Three-column table read by rows: triples (i,j,k) in order sorted from the left.

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A360010 First part of the n-th weakly decreasing triple of positive integers sorted lexicographically. Each n > 0 is repeated A000217(n) times.

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A070770 b + c + d where b >= c >= d >= 0 ordered by b then c then d.

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A194849 Write n = C(i,3)+C(j,2)+C(k,1) with i>j>k>=0; let L[n] = [i,j,k]; sequence gives list of triples L[n], n >= 0.

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A360573 Odd numbers with exactly three zeros in their binary expansion.

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Programs

Maple

Mathematica

PARI

Python

Python

Python

Python

Formula

A360240 Weakly decreasing triples of positive integers sorted lexicographically and concatenated.

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Mathematica

Python

Formula